By corollary 1.1.10 in [1]} \(U^{\prime }_F\in RV_{\gamma -1}\) , hence for some slowly varying function \(L\)  we can write \(U^{\prime }_F(x)=x^{\gamma -1}L(x)\)  (be lemma REF ). Then
\(c_N= \frac{N}{k}U_F^{\prime }\left(\frac{N}{k} \right)= \left(\frac{N}{k}\right)^{\gamma }L\left(\frac{N}{k} \right) .\) 
